Polyomino non-tiling problem: Definition: A p-dimensional n-omino X is a union of integer lattice cubes (of size 1^p) such that i) The interior of X is connected, AND ii) X is topologically equivalent to a closed p-dimensional disk. [Note: I'm adding the nonstandard condition ii) in order to avoid trivial cases of non-tiling due to holes, such as the heptomino with its squares' centers arranged thus: * * * * * * * .] In p-dimensional Euclidean space, find the least N for which there exists a p-dimensional N-omino, copies of which cannot tile p-space. (E.g., it's known that F(2) = 7.) Call this N by the notation F(p) QUESTION I): Can F(p) be determined explicitly for all p ??? ------------- QUESTION II): In any case, can an asymptotic formulas be found ------------- for F(p), as p -> oo ??? --Dan